NUMERICAL STUDY OF QUANTIZED VORTEX INTERACTIONS IN THE NONLINEAR SCHRÖDINGER EQUATION ON BOUNDED DOMAINS

Transcription

1 MULTISCALE MODEL. SIMUL. Vol. 2, No. 2, pp c 24 Societ for Industrial and Applied Mathematics NUMERICAL STUDY OF QUANTIZED VORTEX INTERACTIONS IN THE NONLINEAR SCHRÖDINGER EQUATION ON BOUNDED DOMAINS WEIZHU BAO AND QINGLIN TANG Abstract. In this paper, we stud numericall quantized vortex dnamics and their interactions in the two-dimensional (2D) nonlinear Schrödinger equation (NLSE) with a dimensionless parameter > proportional to the size of the vortex core on bounded domains under either a Dirichlet or a homogeneous Neumann boundar condition (BC). We begin with a review of the reduced dnamical laws for time evolution of quantized vortex centers and show how to solve these nonlinear ordinar differential equations numericall. Then we outline some efficient and accurate numerical methods for discretizing the NLSE on either a rectangle or a disk under either Dirichlet or homogeneous Neumann boundar condition. Based on these efficient and accurate numerical methods for NLSE and the reduced dnamical laws, we simulate quantized vortex interactions of NLSE with different and different initial setups including single vortex, vortex pair, vortex dipole, and vortex cluster, compare them with those obtained from the corresponding reduced dnamical laws, and examine the validit of the reduced dnamical laws. Finall, we investigate radiation and generation of sound waves as well as their impact on vortex interactions in the NLSE dnamics. Ke words. nonlinear Schrödinger equation, quantized vortex, Dirichlet boundar condition, homogeneous Neumann boundar condition, reduced dnamical laws, time splitting AMS subject classifications. 35Q55, 65M6, 65M7, 8Q5, 82D5 DOI..37/ Introduction. Vortices are those waves that possess phase singularities (topological defect) and rotational flows around the singular points. The arise in man phsical areas of different scales and in nature, ranging from liquid crstals and superfluids to nonequilibrium patterns and cosmic strings [4, 37]. Quantized vortices in two dimensions are those particle-like vortices whose centers are zeros of the order parameter, possessing localized phase singularit with the topological charge (also known as winding number, index, or circulation) being quantized. The have been widel observed in man different phsical sstems, such as liquid Helium, tpe-ii superconductors, atomic gases, and nonlinear optics [, 5, 3, 24, 3]. Quantized vortices are ke signatures of superconductivit and superfluidit. Their stud remains one of the most important and fundamental problems since the were predicted b Lars Onsager in 947 in connection with superfluid Helium. In this paper, we consider and stud numericall quantized vortex dnamics and interactions in the two-dimensional (2D) nonlinear Schrödinger equation (NLSE), also known as the Gross Pitaevskii equation (GPE), which is a fundamental equation for modeling and understanding superfluids [2, 6, 7, 9, 38]: (.) i t ψ (x,t)=δψ + 2 ( ψ 2 )ψ, x Ω, t >, Received b the editors Januar 8, 23; accepted for publication (in revised form) Januar 2, 24; published electronicall April, 24. This research was supported b Singapore A*STAR SERC Complex Sstems Research Programme grant Department of Mathematics and Center for Computational Science and Engineering, National Universit of Singapore, Singapore 976, Singapore (matbaowz@nus.edu.sg, nus.edu.sg/ bao/; g888@nus.edu.sg). 4

2 42 WEIZHU BAO AND QINGLIN TANG with initial condition (.2) ψ (x, ) = ψ (x), x Ω, and either a Dirichlet boundar condition (BC) (.3) ψ (x,t)=g(x) =e iω(x), x Ω, t, or a homogeneous Neumann BC ψ (x,t) (.4) =, x Ω, t. n Here, Ω R 2 is a simpl connected and bounded domain, t is time, x =(x, ) R 2 is the Cartesian coordinate vector, ψ := ψ (x,t) is a complex-valued function describing the order parameter for a superfluid, ω is a given real-valued function, ψ and g are given smooth and complex-valued functions satisfing the compatibilit condition ψ (x) =g(x) forx Ω, n =(n,n 2 )andn =( n 2,n ) R 2 satisfing n = n 2 + n 2 2 = are the outward normal and tangent vectors along Ω, respectivel, and > is a given dimensionless constant. Denote the Gross Pitaevskii (GP) or Ginzburg Landau (GL) functional ( energ ) as [, 8, 25] [ (.5) E (t) := ψ (x,t) 2 + ( ψ 2 2 (x,t) 2 ) ] 2 dx = Ekin (t)+e int (t); Ω then it is eas to see that, for the NLSE (.) with either Dirichlet BC (.3) or homogeneous Neumann BC (.4) for general domain Ω, or periodic BC when Ω is a rectangle, the GP functional E (t) is conserved, i.e., E (t) E () for t. Several analtical and numerical studies have dealt with quantized vortex states of the NLSE (.) and their interactions in the whole space R 2 or on bounded domains under different scalings with regard to the distances between different vortices. Based on a formal analsis, Fetter [5] predicted that, to the leading order, the dnamics of vortices in the NLSE (.) would be governed b the same law as that in the ideal incompressible fluid. The same prediction was then given mathematicall b Neu [3] using the method of matched asmptotics. In Neu s work [3], he found vortex states of the NLSE (.) in the whole space R 2 with = for superfluidit and conjectured the stabilit of these states under NLSE dnamics as an open problem [3]. Based on his conjecture on the stabilit, he obtained formall the reduced dnamical laws governing the motion of the vortex centers under the assumption that these vortices are distinct and well-separated; i.e., the reduced dnamical laws are asmptoticall valid when the distances between vortex centers become larger and larger [3]. Under this scaling, the vortex core size is O(). Based on the reduced dnamical law, which is a set of ordinar differential equations (ODEs) for the vortex centers, one can obtain that two vortices with opposite winding numbers (vortex dipole or vortex-antivortex) move in parallel, while the rotate along a circle if the have the same winding number (vortex pair). However, these ODEs are correct onl up to the leading order. Corrections to this leading order approximation due to radiation and/or related questions when long-time dnamics of vortices is considered still remain an important open problem. Using the method of effective action and geometric solvabilit, Ovichinnikov and Sigal confirmed Neu s approximation and derived some leading radiative corrections [34, 35] based on the assumption that the vortices are well-separated, which was

3 QUANTIZED VORTEX IN NLSE ON BOUNDED DOMAINS 43 extended b Lange and Schroers [23] to the stud of the dnamics of overlapping vortices. Recentl, b proposing efficient and accurate numerical methods for discretizing the NLSE in the whole space, Zhang, Bao, and Du [43, 44] compared the dnamics of quantized vortices from the reduced dnamical laws obtained b Neu with those from the direct numerical simulation results of the NLSE under different parameters and/or initial setups. The solved numericall Neu s open problem on the stabilit of vortex states under the NLSE dnamics: vortices with winding number m = ± are dnamicall stable and, respectivel, m > dnamicall unstable [43, 44], which agrees with those conclusions derived asmptoticall b Ovchinnikov and Sigal [33]. In addition, the identified numericall different parameter regimes that the dnamics and interactions of vortex centers obtained from the reduced dnamical laws agree qualitativel and/or quantitativel as well as fail to agree with those obtained from the NLSE dnamics [43, 44]. We remark here that Kevrekidis and his collaborators [8, 2, 28, 3, 36, 4, 42] recentl carried out some ver interesting works on the dnamics and structure stabilit of vortex clusters with several vortices in trapped Bose Einstein condensates (BEC) b studing the GPE with confinement potentials in the whole space R 2. Due to the confinement potential and normalization condition, the situation there is different from ours: (i) the profile of each single vortex is different the densit outside the vortex core (with size at O()) is uniform with value up to the domain boundar in our case (cf. Figure ) and it is zero outside the Thomas Fermi radius in the BEC setup due to the harmonic confinement potential; (ii) in the strong interaction regime, i.e., <, the densit is almost everwhere except the vortex core in our case, while it is at O( ) in the BEC setup when the dimensionless interaction strength in front of the cubic interaction is at O(/ 2 ) [4]; (iii) there is no new vortex generated from the boundar during the dnamics due to the nonzero boundar condition in our case, while there are new vortices generated from the boundar in the BEC setup [2] due to the harmonic trapping potential. Thus the setups there are quite different from that in this paper. Inspired b Neu s work, man researchers have published papers on the stud of the vortex states and dnamics governed b the NLSE (.) on a bounded domain Ω R 2 with the introduction of a small dimensionless parameter <<, which is proportional to the core size of a vortex. Under this scaling, the core size of each vortex is O() and the distances between vortex centers initiall are O(). Mironescu [29] investigated stabilit of the vortices in (.) with (.3) and showed that for fixed winding number m, avortexwith m = is alwas dnamicall stable, while for those of winding number m >, there exists a critical c m such that if > c m, the vortex is stable; otherwise it is unstable. Mironescu s results were then improved b Lin [27] using the spectrum of a linearized operator. Subsequentl, Lin and Xin [25] studied the vortex dnamics on a bounded domain with either a Dirichlet or a Neumann BC, which was further investigated b Jerrard and Spirn [8] using different methods. In addition, Colliander and Jerrard [, 2] studied the vortex structures and dnamics on a torus under a periodic BC. In these studies, the authors derived the reduced dnamical laws which govern the dnamics of vortex centers under the NLSE dnamics when with fixed initial distances between different vortex centers. The obtained that, to the leading order, the vortices move according to the Kirchhoff law in the bounded domain case. However, these reduced dnamical laws cannot indicate radiation and/or sound propagations created b highl corotating or overlapping vortices. It remains a ver fascinating and fundamental problem to understand the vortex-sound interactions [, 32] and how the sound waves modif the motion of vortices [6].

4 44 WEIZHU BAO AND QINGLIN TANG Ver recentl, the authors designed some efficient and accurate numerical methods for studing vortex dnamics and interactions in the GL equation on bounded domains with either a Dirichlet or a Neumann BC [6]. These numerical methods can be extended and applied to stud the rich and complicated phenomena related to vortex dnamics in the NLSE (.) with either Dirichlet BC (.3) or homogeneous Neumann BC (.4) on bounded domains. The main goals of this paper are (i) to present efficient and accurate numerical methods for discretizing the reduced dnamical laws and the NLSE (.) on bounded domains under different BCs, (ii) to understand numericall how the BC and radiation, as well as geometr of the domain, affect vortex dnamics and interactions, (iii) to stud numericall the vortex interactions in the NLSE dnamics and/or compare them with those from the reduced dnamical laws under different initial setups and parameter regimes, and (iv) to identif cases where the vortex dnamics and interactions from the reduced dnamical laws agree/disagree qualitativel and/or quantitativel with those from the NLSE dnamics. The rest of the paper is organized as follows. In section 2, we briefl review the reduced dnamical laws of vortex interactions under the NLSE (.) with either a Dirichlet or a homogeneous Neumann BC and present numerical methods to discretize them. In section 3, efficient and accurate numerical methods are briefl outlined for discretizing the NLSE on bounded domains with different BCs. In section 4, numerical results are reported for vortex interactions of the NLSE under the Dirichlet BC, and similar results for the NLSE under the homogeneous Neumann BC are reported in section 5. Finall, some conclusions are drawn in section The reduced dnamical laws and their discretization. In this section, we review two different forms of the reduced dnamical laws for the dnamics of vortex centers in the NLSE (.) with either a Dirichlet or a homogeneous Neumann BC, show their equivalence, and present efficient numerical methods to discretize them. We assume that, in the initial data ψ,thereareexactlm isolated and distinct vortices whose centers are located at x =(x, ), x 2 =(x 2, 2 ),..., x M =(x M, M ) with winding numbers n, n 2,...,n M, respectivel. The winding number of each vortex can be chosen as either or, i.e., n j =or forj =, 2,...,M. At time t, the M isolated and distinct vortex centers are located at x (t) =(x (t), (t)), x 2 (t) =(x 2 (t), 2 (t)),...,andx M (t) =(x M (t), M (t)). Denote X := (x, x 2,...,x M ), X := X(t) =(x (t), x 2 (t),...,x M (t)), t ; the renormalized energ associated with the M vortex centers is defined as [9, 24] (2.) W (X) :=W (X(t)) = n j n l ln x j (t) x l (t), t. j l M 2.. Under the Dirichlet BC. For the NLSE (.) with Dirichlet BC (.3), when, two different forms of the reduced dnamical laws have been obtained in the literature for the motion of the M vortex centers. The first, which is widel used, has been derived formall and rigorousl in the literature; see, for instance, [9,, 22, 26] and references therein: (2.2) ẋ j (t) = n j J xj [W (X)+W dbc (X)], j =,...,M, t>, with initial condition (2.3) x j () = x j, j =, 2,...,M.

5 QUANTIZED VORTEX IN NLSE ON BOUNDED DOMAINS 45 Here, J is a 2 2 smplectic matrix defined as ( J = and the renormalized energ W dbc (X) comes from the effect of the Dirichlet BC associated with the M vortex centers X = X(t) and is defined as [9, 24] M M W dbc (X) = n j R(x j ; X)+ R(x; X) + n j ln x x j n ω(x) ds, Ω 2π j= where, for an fixed X Ω M, R(x; X) is a harmonic function in x, i.e., ), j= (2.4) ΔR(x; X) =, x Ω, with the Neumann BC (2.5) R(x; X) n = n ω(x) n M n l ln x x l, l= Using an identit in [9, eq. (5), p. 84], xj [W (X)+W dbc (X)] = 2n j x R(x; X) + M l=&l j x Ω. n l ln x x l x=xj, the reduced dnamical law (2.2) can be simplified, for j M, to (2.6) ẋ j (t) =2J x R (x; X) x=xj(t) + M x j (t) x l (t) n l x j (t) x l (t) 2, t >. l=&l j The second form of the reduced dnamical law was obtained b Lin and Xin [25] for j M as (2.7) ẋ j (t) =2J J x H (x; X) x=xj(t) + M x j (t) x l (t) n l x j (t) x l (t) 2, t >, l=&l j where for an fixed X Ω M, H(x; X) is a harmonic function in x and satisfies the BC (2.8) H(x; X) n = n ω(x) n M n l ln x x l, l= x Ω. In the above two different forms of the reduced dnamical laws for the NLSE, although the two harmonic functions R(x; X) andh(x; X) satisfdifferentbcs,the are equivalent. In fact, the are both equivalent to the following reduced dnamical law: For j M, (2.9) ẋ j (t) =2J J x Q (x; X) x=xj(t) + M x j (t) x l (t) n l x j (t) x l (t) 2, t >, l=&l j

6 46 WEIZHU BAO AND QINGLIN TANG where, for an fixed X Ω M, Q(x; X) is a harmonic function in x and satisfies the Dirichlet BC M (2.) Q(x; X) =ω(x) n l θ(x x l ), l= x Ω, with the function θ : R 2 [, 2π) being defined as (2.) cos(θ(x)) = x x, sin(θ(x)) = x, x =(x, ) R2. Lemma 2.. For an fixed X Ω M, we have the identit (2.2) J x Q (x; X) = x R (x; X) =J x H (x; X), x Ω, which immediatel implies the equivalence of the three reduced dnamical laws (2.6), (2.7), and(2.9). Proof. For an fixed X Ω M,sinceQis a harmonic function, there exists a function ϕ (x) such that Thus, ϕ (x) satisfies the Laplace equation J x Q (x; X) = ϕ (x), x Ω. (2.3) Δϕ (x) = (J x Q(x; X)) = x ϕ (x) x ϕ (x) =, x Ω, with the Neumann BC (2.4) n ϕ (x) =(J x Q(x; X)) n = x Q(x; X) n = n Q(x; X), x Ω. Noting (2.), we obtain for x Ω n ϕ (x) = n ω(x) n M l= n l θ(x x l )= n ω(x) n M n l ln x x l. l= Combining the above equalit with (2.3), (2.4), and (2.5), we get Thus Δ(R(x; X) ϕ (x)) =, x Ω; n (R(x; X) ϕ (x)) =, x Ω. R(x; X) =ϕ (x)+constant, x Ω, which immediatel implies the first equalit in (2.2). The second equalit in (2.2) can be proved in a similar wa, but we omit it here for brevit Under the homogeneous Neumann BC. Similarl, for the NLSE (.) with the homogeneous Neumann BC (.4), when, there are also two different forms of the reduced dnamical laws for the motion of the M vortex centers. Again, b introducing the renormalized energ W nbc, which comes from the effect of the homogeneous Neumann BC associated with the M vortex centers X = X(t), (2.5) W nbc (X) = M n j R(xj ; X), j=

7 QUANTIZED VORTEX IN NLSE ON BOUNDED DOMAINS 47 where, for an fixed X Ω M, R(x; X) is a harmonic function in x and satisfies the Dirichlet BC (2.6) R(x; M X) = n l ln x x l, l= x Ω, the first form has been derived formall and rigorousl b several authors in the literature [, 8, 22] as (2.7) ẋ j (t) = n j J xj [W (X)+W nbc (X)], j =,...,M, t>. Using the identit (2.8) xj [W (X)+W nbc (X)] = 2n j x R(x; X)+ M l=&l j the above reduced dnamical laws collapse for j M as (2.9) ẋ j (t) =2J x R (x; X) x=xj(t) + M l=&l j n l n l ln x x l xj, x j (t) x l (t) x j (t) x l (t) 2, t >. Similarl, the second form of the reduced dnamical law was obtained b Lin and Xin [25] for j M as (2.2) ẋ j (t) =2J J x Q (x; X) x=xj(t) + M l=&l j n l x j (t) x l (t) x j (t) x l (t) 2, t >, where, for an fixed X Ω M, Q(x; X) is a harmonic function in x and satisfies the Neumann BC (2.2) Q(x; X) n = n M n l θ(x x l ), l= x Ω. Similarl to the proof of Lemma 2., we can establish the equivalence of the above two different forms of the reduced dnamical laws. Lemma 2.2. The reduced dnamical laws (2.9) and (2.2) are equivalent. In order to compare the solution of the reduced dnamical laws (2.6) or (2.9) and (2.9) or (2.2) with those from the NLSE under a Dirichlet or a homogeneous Neumann BC, respectivel, the ODEs (2.6) or (2.9) and (2.9) or (2.2) are discretized b the standard second order leap-frop method or fourth order Runge Kutta method. For each fixed X Ω M, when the domain Ω is a rectangle, the Laplace equation (2.4) with BC (2.5) or (2.) or (2.6) or (2.2) is discretized b the standard second order finite difference method, and, respectivel, when the domain Ω is a disk, the are discretized in the θ-direction via the Fourier pseudospectral method and in the r- direction via the finite element method (FEM) with (r, θ) the polar coordinates. For details, we refer the reader to [6] but omit them here for brevit.

8 48 WEIZHU BAO AND QINGLIN TANG 3. Numerical methods. In this section, we outline briefl some efficient and accurate numerical methods for discretizing the NLSE (.) in either a rectangle or a disk with initial condition (.2) and either a Dirichlet or a homogeneous Neumann BC. The ke ideas in our numerical methods are based on (i) appling a time-splitting method, which has been widel used for nonlinear partial differential equations (PDEs) [4] to decouple the nonlinearit in the NLSE [5, 7, 43], and (ii) adapting a proper finite difference method and/or FEM to discretize a free Schrödinger equation [5, 6]. Let τ := t > be the time step size, and denote t n = nτ for n. For n =,,..., from time t = t n to t = t n+, the NLSE (.) is solved in two splitting steps. One first solves (3.) i t ψ (x,t)= 2 ( ψ 2 )ψ, x Ω, t t n, for the time step of length τ, followed b solving (3.2) i t ψ (x,t)=δψ, x Ω, t t n, for the same time step. The discretization of (3.2) will be outlined later. For t [t n,t n+ ], we can easil obtain the following ODE for ρ(x,t)= ψ (x,t) 2 : (3.3) t ρ(x,t)=, t t n, x Ω, which implies that (3.4) ρ(x,t)=ρ(x,t n ), t t n, x Ω. Plugging (3.4) into (3.), we can integrate it exactl to get [ (3.5) ψ (x,t)=ψ (x,t n )exp ] 2 ( ψ (x,t n ) 2 )(t t n ), t t n, x Ω. We remark here that, in practice, we alwas use the second order Strang splitting [4]; that is, from time t = t n to t = t n+, (i) evolve (3.) for half time step τ/2 with initial data given at t = t n ; (ii) evolve (3.2) for one step τ starting with the new data; and (iii) evolve (3.) for half time step τ/2 again with the newer data. When Ω = [a, b] [c, d] is a rectangular domain, we denote h x = b a N and h = d c L, with N and L being two even positive integers as the mesh sizes in the x-direction and the -direction, respectivel. Similarl to the discretization of the gradient flow with constant coefficient [6], when the Dirichlet BC (.3) is used for (3.2), it can be discretized b using the fourth order compact finite difference discretization for spatial derivatives followed b a Crank Nicolson finite difference (CNFD) scheme for temporal derivatives [6], and when homogeneous Neumann BC (.4) is used for (3.2), it can be discretized b using cosine spectral discretization for spatial derivatives followed b integrating in time exactl [6]. The details are omitted here for brevit. Combining the CNFD and cosine pseudospectral discretization for Dirichlet and homogeneous Neumann BCs, respectivel, with the second order Strang splitting, we can obtain time-splitting Crank Nicolson finite difference (TSCNFD) and time-splitting cosine pseudospectral (TSCP) discretizations for the NLSE (.) on a rectangle with Dirichlet BC (.3) and homogeneous Neumann BC (.4), respectivel. Both TSCNFD and TSCP discretizations are unconditionall stable, second order in time, and have memor cost O(NL) and computational cost O (NLln(NL)) per time step. In addition, TSCNFD is fourth order in space and TSCP is spectral order in space.

9 QUANTIZED VORTEX IN NLSE ON BOUNDED DOMAINS 49 When Ω = {x x <R} := B R () isadiskwithr > a fixed constant, similarl to the discretization of the GPE with an angular momentum rotation [3, 5, 43] and/or the gradient flow with constant coefficient [6], it is natural to adopt the polar coordinate (r, θ) in the numerical discretization b using the standard Fourier pseudospectral method in the θ-direction [39], FEM in the r-direction, and the Crank Nicolson method in time [3, 5, 43]. Again, the details are omitted here for brevit. 4. Numerical results under Dirichlet BC. In this section, we report numerical results for vortex interactions of the NLSE (.) under Dirichlet BC (.3) and compare them with those obtained from the reduced dnamical laws (2.6) with (2.3). We stud how the dimensionless parameter, initial setup, boundar value, and geometr of the domain Ω affect the dnamics and interactions of vortices. For a given bounded domain Ω, the NLSE (.) is unchanged b the rescaling x dx, t d 2 t, and d with d the diameter of Ω. Thus without lose of generalit, hereafter, without specification, we alwas assume that the diameter of Ω is O(). The initial condition ψ in (.2) is chosen as (4.) ψ (x) =ψ (x, ) M =eih(x) φ n j (x x j ), j= x =(x, ) Ω, where M>isthe total number of vortices in the initial data, h(x) is a real-valued harmonic function corresponding to phase in-painting at t =,θ(x) is defined in (2.), and for j =, 2,...,M, n j =or, and x j =(x j, j ) Ω are the winding number and initial location of the jth vortex, respectivel. In addition, Here, f (r) ischosenas (4.2) f (r) = φ n j (x) =f ( x ) e injθ(x), x =(x, ) Ω. {, r R =.25, fv(r), r R, where f v is the solution of the following problem: [ r d dr ( r d ) dr r 2 + ( (f 2 v (r)) 2)] fv (r) =, <r<r, with the Dirichlet BC f v (r =)=, f v (r = R )=. The solution fv of the above problem is computed numericall, and we depict the function f (r) in Figure with different. Thus in the initial data (4.), there are exactl M distinct vortices with core size at O() under a phase in-painting h(x). The function g in the Dirichlet BC (.3) is chosen as (4.3) g(x) =e i(h(x)+ M j= njθ(x x j )), x Ω, so that it is compatible with ψ at t =.

10 42 WEIZHU BAO AND QINGLIN TANG =/25 =/5 =/ f (r) r Fig.. Plot of the function f (r) in (4.2) with different. In addition, in the following sections, we mainl consider the following six different modes of the phase in-painting h(x): Mode : h(x) =. Mode : h(x) =x +. Mode 2: h(x) =x. Mode 3: h(x) =x 2 2. Mode 4: h(x) =x x. Mode 5: h(x) =x 2 2 2x. To simplif our presentation, for j =, 2,...,M, hereafter we let x j (t) and (t) bethejth vortex center in the NLSE dnamics and the corresponding reduced x r j dnamical laws, respectivel, and denote d j (t) = x j (t) xr j (t) as their difference. Furthermore, in the presentation of the figures, the initial location of a vortex with winding number +, and the location where two vortices merge are marked as +,, and, respectivel. Finall, in our computations, if not specified, we take Ω = [, ] 2 in (.), mesh sizes h x = h =,andtimestepτ = 6. The NLSE (.) with (.3), (.2), and (4.) is solved b the method TSCNFD presented in section Single vortex. Here we present numerical results of the motion of a single quantized vortex in the NLSE (.) dnamics and the corresponding reduced dnamics; i.e., we take M =,n = in (4.). To stud how the initial phase shift h(x) and initial location of the vortex x affect the motion of the vortex and to understand the validit of the reduced dnamical law, we consider the following cases: Cases I III: x =(, ), and h(x) is chosen as Mode, 2, and 3, respectivel. Cases IV VIII: x =(., ), and h(x) ischosenasmode,2,3,4,and5, respectivel. Cases IX XI: x =(.,.), and h(x) is chosen as Mode 3, 4, and 5, respectivel. Moreover, to stud the effect of domain geometr, we consider Ω of three tpes: Tpe I, a square Ω = [, ] 2 ; Tpe II, a rectangle Ω = [, ] [.65,.65]; and Tpe III, a unit disk Ω = B (). Thus we also stud the following four additional cases: Cases XII XIII: x =(, ), h(x) =x +, and Ω is chosen as Tpe II and III, respectivel. Cases XIV XV: x =(., ), h(x) =x2 2, and Ω is chosen as Tpe II and III, respectivel. Figure 2 depicts the trajector of the vortex center when = 4 in (.) for Cases I VI and d with different for Cases I, V, and VI. Figure 3 shows the trajector of the vortex center when = 64 in (.) for Cases VI XI, while Figure 4 shows that for Cases XII XVII when = 32 in (.). From Figures 2 4 and additional numerical experiments (not shown here for brevit), we can draw the following conclusions:

11 QUANTIZED VORTEX IN NLSE ON BOUNDED DOMAINS 42 x x x x.34 x.4 x.2 d.7 =/32 =/5 =/8 d.2 =/32 =/5 =/8 d.6 =/6 =/4 =/8.5 t.5 t.8 t.6 Fig. 2. Trajector of the vortex center in NLSE under Dirichlet BC when = for Cases 4 I VI (from left to right and then from top to bottom in the top two rows), and d for different for Cases I, V, andvi (from left to right in bottom row) in section 4.. x x x x x x Fig. 3. Trajector of the vortex center in NLSE dnamics under Dirichlet BC when = 64 (solid line) and from the reduced dnamical laws (dashed line) for Cases VI XI (from left to right and then from top to bottom) in section 4.. (i) When h(x), the vortex center does not move, and this is similar to the case in the whole space. (ii) When h(x) = (x + b)(x b ) with b, the vortex does not move if x =(, ), while it does move if x (, ) (cf. Figure 2, Cases III and VI for b =).

12 422 WEIZHU BAO AND QINGLIN TANG.65 x.65 x.65 x.65 x Fig. 4. Trajector of the vortex center in the NLSE under the Dirichlet BC when = 4 for Cases I, XII, XIII, VI, XIV, andxv (from left to right and then from top to bottom) in section 4.. (iii) When h(x) andh(x) (x + b)(x b ) with b, in general, the vortex center does move. For the NLSE dnamics, there exists a critical value c depending on h(x), x, and Ω such that if < c, the vortex will move periodicall in a closed loop (cf. Figure 2); otherwise its trajector will not be a closed loop. This differs significantl from the situation in the reduced dnamics, where the trajector is alwas periodic (cf. Figure 3, dashed line). Thus the reduced dnamical laws fail qualitativel when > c. It should be an interesting problem to find out how this critical value depends on h(x), x, and the geometr of Ω. (iv) In general, the initial location, the geometr of the domain, and the boundar value will all affect the motion of the vortex (cf. Figure 4). (v) When, the dnamics of the vortex center under the NLSE dnamics converges uniforml in time to that of the vortex center under the reduced dnamics (cf. Figure 2, bottom row). This verifies numericall the validation of the reduced dnamical laws Vortex pair. Here we present numerical results of the interactions of a vortex pair under the NLSE (.) dnamics and its corresponding reduced dnamical laws; i.e., we take M =2,n = n 2 =,x 2 = x =(d, ) with <d < (i.e., the two vortices are initiall located smmetricall on the x-axis), h(x) in (4.), and = 4 in (.). Figure 5 depicts the trajector of the vortex pair and time evolution of E (t), Ekin (t), x (t), x 2(t), and d (t) fordifferentd. From Figure 5 and additional numerical results (not shown here for brevit), we can draw the following conclusions for the interaction of a vortex pair under NLSE dnamics (.) with a Dirichlet BC: (i) The total energ is conserved numericall ver well during the dnamics. (ii) The initial locations of the two vortices affect significantl the motion of the vortex pair. If the vortex pair is smmetric initiall, then the move periodicall and their trajectories are smmetric, i.e., x (t) = x 2 (t) fort. Furthermore, for both the reduced dnamical law and NLSE dnamics, there exist critical values, sa d r c and d c, respectivel, such that if d <d r c (or d <d c in the NLSE dnamics), the two vortices will rotate with each other and move along a circle-like trajector; otherwise,

13 QUANTIZED VORTEX IN NLSE ON BOUNDED DOMAINS x or x t E kin E d.2. =/4 =/5 =/ x.8 or 2 x or x2..75 t.5 2 t t E kin E.94 d t.5 =/6 =/25 =/4 x or 2 2 t t 4 2 t 4 Fig. 5. Trajector of the vortex pair (left column), time evolution of x (t) and x 2 (t) (second column), E (t) and Ekin (t) (third column), and d (t) (last column) for d =. (top row) and d =.5 (bottom row) in section d c dc r. ln(d c dc r ) ln() Fig. 6. Critical value d c for the interaction of a vortex pair of the NLSE (.) under the Dirichlet BC with h(x) =in (4.) for different in section 4.2. the will move along a crescent-like trajector (cf. Figure 5). We find numericall the critical values d r c.4923 and d c for <<, which are depicted in Figure 6. From these values, we obtain numericall the following relation for d r c and d c : d c = dr c , <<. (iii) For an fixed d, the dnamics of the two vortex centers under the NLSE dnamics converges uniforml in time to that of the two vortex centers under the reduced dnamical laws (cf. Figure 5) when. However, for fixed, the reduced dnamical law fails qualitativel to describe the motion of vortices if d r c <d <d c Vortex dipole. Here we present numerical results of the interactions of a vortex dipole under the NLSE (.) dnamics and its corresponding reduced dnamical laws; i.e., we take M =2,h(x), n = n 2 =, x 2 = x =(d, ) in (4.) with d =.5, and = 25 in (.). Figure 7 depicts contour plots of ψ (x, t) at different times, trajector of the vortex dipole, and time evolution of x (t), x 2(t), and d (t). From Figure 7 and additional numerical results (not shown here for brevit), we can draw the following conclusions: (i) The total energ is conserved numericall ver well during the dnamics. (ii) The vortex dipole under the NLSE dnamics moves upward smmetricall with respect to the -axis, and finall the two vortices in the vortex dipole merge with

14 424 WEIZHU BAO AND QINGLIN TANG.5 x.5.5 x.5 x or x2.6.8 d.6.65 t.33.4 or 2.65 t.33 =/8 =/6 =/32.5 t.3 Fig. 7. Contour plots of ψ (x,t) at different times (top two rows), trajector of x (t) and x 2 (t) in the NLSE dnamics (bottom left) and of xr (t) and xr 2 (t) in reduced dnamics (bottom, second plot), and time evolution of x (t) and x 2 (t) (bottom, third plot) and d (t) (bottom right) for the dnamics of a vortex dipole in section 4.3. each other and the are annihilated somewhere near the top boundar simultaneousl. The distance between the merging place and the boundar is O() when is small. After merging, new waves will be created and reflected b the top boundar. The new waves will then move back into the domain and be reflected back into the domain again when the hit the left, right, and bottom boundaries (cf. Figure 7). Moreover, the two vortices in the vortex dipole in the NLSE dnamics will alwas merge in some place near the top boundar for all d (cf. Figure 7, bottom left). However, in the reduced dnamics, the two vortices never merge inside Ω (cf. Figure 7 bottom, second plot). Hence, the reduced dnamical law fails qualitativel when the vortex dipole is near the boundar. (iii) When, the dnamics of the two vortex centers under the NLSE dnamics converges uniforml to that of the two vortex centers under the reduced dnamical laws (cf. Figure 7) before the two vortices merge with each other or are near the boundar. Again, this verifies numericall the validation of the reduced dnamical laws in this case. In fact, based on our extensive numerical experiments, the motion of the two vortex centers from the reduced dnamical laws agrees with that of the two vortices from the NLSE dnamics qualitativel when << and quantitativel when < when the two vortices are not too close to the boundar Vortex cluster. Here we present numerical studies on the dnamics of vortex clusters in the NLSE (.) with Dirichlet BC (.3); i.e., we choose the initial

15 QUANTIZED VORTEX IN NLSE ON BOUNDED DOMAINS x x.4.4 x.4.9 x (t).9 x (t).9 x (t).6 (t) x 2 (t).6 (t) x 2 (t).6 (t) x (t) (t).3 2 (t).3 2 (t) t t t.2.54 x x x.9 x 3 (t) x 3 (t).4 x (t) (t) x (t) 2 (t) 2.5 (t) 3 x 2 (t) 2 (t).2 (t) t t t Fig. 8. Trajector of the vortex x (solid line), x 2 (dashed-dotted line), and x 3 (dashed line) (first and third rows) and their corresponding time evolution (second and fourth rows) for Case I (top two rows) and Case II (bottom two rows) for small time (left column), intermediate time (middle column), and large time (right column) with = 4 and d =.25 in section 4.4. data (.2) as (4.) and stud the following four cases: Case I. M =3,n = n 2 = n 3 =,x = x 3 =(d, ), and x 2 =(, ). Case II. M =3,n = n 2 = n 3 =,x = x 3 =(d, ), and x 2 =(, ). Case III. M = 4, n = n 2 = n 3 = n 4 =, x = x 2 = (d, ), and x 3 = x 4 =(,d 2)with<d,d 2 <. Case IV. Ω = B 5 (), M =9,n = n 2 = = n 9 =, and the nine vortex centers are initiall located on the 3 3 uniform mesh points for the rectangle [ d,d ] 2 with <d <. Figure 8 depicts the trajector and time evolution of x (t), x 2(t), and x 3(t) inthe NLSE dnamics for Cases I and II. Figure 9 shows contour plots of ψ at different times in the NLSE dnamics for Case III, and Figure depicts contour plots of ψ, as well as slice plots of ψ (x,,t), showing sound wave propagation of the NLSE dnamics in Case IV. Based on Figures 8 and additional computations (not shown here for brevit), we can draw the following conclusions: (i) For Case I, the middle vortex (initiall at the origin) will not move, while the other two vortices rotate clockwise around the origin for some time. This dnamics

16 426 WEIZHU BAO AND QINGLIN TANG Fig. 9. Contour plots of ψ (x,t) with = at different times for the NLSE dnamics of a 6 vortex cluster in Case III with different initial locations: d = d 2 =.25 (top two rows); d =.55, d 2 =.25 (middle two rows); d =.25, d 2 =.55 (bottom two rows) in section 4.4.

17 QUANTIZED VORTEX IN NLSE ON BOUNDED DOMAINS t=.2 t=.57 ψ (x,,t).56 ψ (x,,t).56 5 x 5 5 x 5.2 t=.74.2 t=.29 ψ (x,,t).56 ψ (x,,t).56 5 x 5.8 t= x 5.8 t=.588 ψ (x,,t).54 ψ (x,,t).54 5 x 5.8 t= x 5.8 t=.969!ψ (x,,t).54 ψ (x,,t).54 5 x 5 5 x 5 Fig.. Contour plots of ψ (x,t) (left) and slice plots of ψ (x,,t) (right) at different times showing sound wave propagation under the NLSE dnamics of a vortex cluster in Case IV with d =.5 and = in section agrees ver well with the NLSE dnamics in the whole plane [43, 44]. After some time, the above smmetric structure is broken due to boundar effect and numerical errors; i.e., the middle vortex will begin to move toward one of the other two vortices and form a pair of vortices, then the two vortices in the pair will rotate with each other and the vortex pair will rotate with the leftover single vortex for a while. Then this pair will separate and one of them will form a new vortex pair with the single vortex, while the other becomes a new single vortex rotating with them. This process will be repeated tautologicall like three dancers exchanging their partners alternativel. The dnamical pattern after the smmetric structure breaking depends highl on the mesh size and time step chosen. (ii) For Case II, similarl to Case I, the middle vortex (initiall at the origin) will not move, while the other two vortices rotate counterclockwise around the origin for some time. This dnamics agrees ver well with the NLSE dnamics in the whole plane [43, 44]. After some time, again the above smmetric structure is broken due to boundar effect and numerical errors; i.e., the middle vortex will begin to move toward one of the other two vortices and form a vortex dipole which will move nearl parallel to the boundar and merge near the boundar. Sound waves will be created and reflected back into the domain and will drive the leftover vortex in the domain to move (cf. Figure 8). Again the dnamical pattern after the smmetric structure breaking depends highl on the mesh size and time step chosen. From section 4., we

18 428 WEIZHU BAO AND QINGLIN TANG know that a single vortex in the NLSE with h(x) = does not move, and hence this example illustrates clearl the vortex-sound interactions which cannot be predicted b the reduced dnamical laws since the are valid onl before annihilation and when is ver small. (iii) For Case III, the four vortices form two vortex dipoles when t is small. Then the two dipoles move outward in opposite directions, and finall the two vortices in each vortex dipole merge and are annihilated at some place near the boundar. If d = d 2, the two vortex dipoles move smmetricall with respect to the line = x and the two vortices in each dipole merge some place near the top-right and bottom-left corners, respectivel; if d >d 2, the move to the top boundar and the bottom boundar, respectivel, and again the two vortices in each dipole merge with each other when the are close to the boundar; and if d <d 2, the move to the left boundar and the right boundar, respectivel, and again the two vortices in each dipole merge with each other when the are close to the boundar. New waves are created after the two vortices in each dipole merge with each other and are reflected back into the domain when the hit the boundar (cf. Figure 9). (iv) For Case IV, vortex (initiall at the origin) does not move due to smmetr, while the other eight vortices rotate clockwise and move along two circular trajectories (cf. Figure ). During the dnamics, sound waves are generated and the propagate outward, and are reflected back into the domain when the hit the boundar. The distances between the one centered at the origin and the other eight vortices increase when sound waves are radiated outward; on the other hand, the decrease and become even smaller than their initial distances when sound waves are reflected b the boundar and move back into the domain (cf. Figure ). This example clearl shows the impact of sound waves on the dnamics of vortices. Similarl to the case in BEC setups [2, 28, 36, 42], the smmetric structure near t = in Cases I and II for three vortices is dnamicall unstable; i.e., smmetric structure breaking will happen ver quickl if we perturb the location of either the central vortex or one of the side vortices a little bit. To show this, we perturb the initial locations of some vortices in Case I to the following: Tpe I. Asmmetric perturbation: x 2 from (, ) (δ, ) with δ>. Tpe II. Asmmetric perturbation: x from (d, ) (d δ, ) with δ>. Tpe III. Smmetric perturbation: x from (d, ) (d δ, ) and x 3 from ( d, ) ( d + δ, ) with δ>. Figure depicts the trajector of x (t), x 2 (t), and x 3 (t)aswellasd 2 (t) := x 2 (t) for the above three perturbations. From this figure and additional computations (not shown here for brevit), we see that the smmetric structure is dnamicall stable under a smmetric perturbation, and it is dnamicall unstable under a small asmmetric perturbation. For more stabilit analsis of vortex clusters, we refer the reader to [8, 2, 28, 3, 36, 4, 42] and references therein Radiation and sound wave. Here we stud numericall how the radiation and sound waves affect the dnamics of quantized vortices in the NLSE dnamics under the Dirichlet BC. To this end, we consider two tpes of perturbation. Tpe I: Perturbation on the initial data; i.e., we take the initial data (.2) as (4.4) ψ (x, ) = ψ δ, (x) =ψ (x)+δe (( x.8)2 + 2), x =(x, ) Ω, where ψ is given as in (4.) with h(x), M =2,n = n 2 =,andx = x 2 = (., ); i.e., we perturb the initial data for studing the interactions of a vortex pair b a Gaussian function with amplitude δ. Thenwetakeδ = and let go to and

19 QUANTIZED VORTEX IN NLSE ON BOUNDED DOMAINS trajector for t [,.].3 trajector for t [,.] trajector for t [,.52] x x x.3.28 x 2(t).4 δ=.4 δ=.2 δ=. δ=.5.28 x 2 (t).4 δ=.4 δ=.2 δ=. δ=.5 x 2(t).3.5 δ=.5 δ=.4 δ=.2 δ=.5.6 t.2.6 t.2.29 t.58 Fig.. Trajector of the vortex centers x (t) (dashed line), x 2 (t) (solid line), and x 3 (t) (dashed-dotted line) with δ =.5 (top row) and time evolution of the distance of x 2 to the origin, i.e., x 2 (t) for different δ (bottom row) under the NLSE dnamics with = 4 and d =.25 for perturbation Tpe I (first column), Tpe II (second column), and Tpe III (third column) in section 4.4. d δ,.6.3 δ=, =/64 δ=, =/8 δ=, =/ d δ,.6.8 δ==/64 δ==/8 δ==/.5 t..5 t. Fig. 2. Time evolution of d δ, (t) for nonperturbed initial data (left) and perturbed initial data (right) in section 4.5. solve the NLSE (.) with initial condition (4.4) for the vortex centers x δ, (t) and (t) and compare them with those from the reduced dnamical law. We denote x δ, 2 d δ, j (t) = x δ, j (t) x r j (t) for j =, 2 as the error. Figure 2 depicts time evolution of d δ, (t) for the case when δ =, i.e., small perturbation, and the case when δ =, i.e., no perturbation. From this figure, we can see that the dnamics of the two vortex centers under the NLSE dnamics converges to that of the two vortex centers obtained from the reduced dnamical law when without perturbation (cf. Figure 2, left). On the contrar, the two vortex centers under the NLSE dnamics do not converge to those obtained from the reduced dnamical law when with small perturbation (cf. Figure 2, right). This clearl demonstrates radiation and sound wave effects on vortices in the NLSE dnamics with the Dirichlet BC. Tpe II: Perturbation b an external potential; i.e., we replace the uniform

20 43 WEIZHU BAO AND QINGLIN TANG.8 t=.8 t=.22 ψ (x,,t).54 ψ (x,,t).54 5 x 5.8 t=.52 5 x 5.8 t=.6 ψ (x,,t).54 ψ (x,,t).54 5 x 5 5 x 5.8 t=..8 t=.56 ψ (x,,t).54 ψ (x,,t).59 5 x 5 5 x 5.8 t= t=2.97 ψ (x,,t).59 ψ (x,,t).63 5 x 5 5 x 5 Fig. 3. Surface plots of ψ (x,t) (left) and slice plots of ψ (x,,t) (right) at different times showing sound wave propagation under the NLSE dnamics in a disk with = and a perturbation 4 in the potential in section 4.5. potential in (.) b a nonuniform potential and solve the NLSE (4.5) i t ψ (x,t)=δψ + 2 ( W (x,t) ψ 2 )ψ, x Ω, t >, with (4.6) W (x,t)= { sin(2t) 2, t [,.5],, t >.5, x Ω. The initial data is chosen as (4.) with M =,n =,x =(, ), Ω = B 5 (), and = 4. In fact, the perturbation is introduced when t [,.5] and is removed after t =.5. Figure 3 illustrates surface plots of ψ and slice plots of ψ (x,,t)at different times showing sound wave propagation. From Figure 3, we can see that the perturbed vortex configuration rotates and radiates sound waves. This agrees well with some former prediction in the whole plane, for example, in Lange and Schroers [23] for the case M = 2. The waves will be reflected back into the domain when the hit the boundar and then be absorbed b the vortex core. Then the

21 QUANTIZED VORTEX IN NLSE ON BOUNDED DOMAINS 43 d.4 =/6 =/25 =/32 d.4 =/8 =/6 =/ x.8 t 3.6 x.8 t 3.6 Fig. 4. Trajector of the vortex center when = 32 and time evolution of d for different for the motion of a single vortex in NLSE under the homogeneous Neumann BC with x =(.35,.4) (left two figures) or x =(,.2) (right two figures) in (5.) in section 5.. vortex core will radiate new waves and the process is repeated tautologicall. This process explicitl illustrates the radiation in the NLSE dnamics. Remark 4.. Basedonthisexampleandothernumerical results (not shown here for brevit), we can conclude that the vortex with winding number m = ± isdnamicall stable under the NLSE dnamics in a bounded domain with a perturbation in the initial data and/or external potential. Meanwhile, we also found numericall that the vortex with winding number m =2and = 32 is also dnamicall stable under a perturbation in the external potential. Actuall, Mironescu [29] indicated that for a vortex with winding number m >, there exists a critical value c m such that if < c m, the vortex is unstable; otherwise the vortex is stable. It was also numericall observed that a vortex with m > is unstable under a perturbation in the potential but stable under a perturbation in the initial data in the whole plane case [43]. Hence, it would be an interesting problem to investigate numericall how the stabilit of a vortex depends on its winding number, value of, and strength and/or tpe of the perturbation under the NLSE dnamics on bounded domains. 5. Numerical results under Neumann BC. In this section, we report numerical results for vortex interactions of the NLSE (.) under the homogeneous Neumann BC (.4) and compare them with those obtained from the reduced dnamical laws (2.9) with (2.3). The initial condition ψ in (.2) is chosen as (5.) ψ (x) =ψ (x, ) M =eihn(x) φ n j (x x j ), j= x =(x, ) Ω, where h n (x) is a harmonic function satisfing the Neumann BC n h n(x) = n M n l θ(x x l ), l= x Ω. In fact, the choice of h n (x) is such that ψ satisfies the homogeneous Neumann BC (.4). The NLSE (.) with (.4), (.2), and (5.) is solved b the TSCP method presented in section Single vortex. Here we present numerical results of the motion of a single quantized vortex under the NLSE (.) dnamics and its corresponding reduced dnamical laws, i.e., we take M =andn = in (5.). Figure 4 depicts trajector of the vortex center for different x in (5.) when = 32 in (.) and d for different. From Figure 4 and additional numerical results (not shown here for brevit), we can see the following:

22 432 WEIZHU BAO AND QINGLIN TANG x or x Ekin.5 t E.5 d.23 or 2 =/8 =/6 =/32 x 2.92 t t t.86 Fig. 5. Trajector of the vortex pair (left), time evolution of E and Ekin (second figure), x (t) and x 2 (t) (third figure), and d (t) (right) in the NLSE dnamics under homogeneous Neumann BC with = 32 and d =.5 in section 5.2. (i) If x =(, ), the vortex will not move for all times; otherwise, the vortex will move, and its initial location x does not affect its motion qualitativel. Actuall, it moves periodicall in a circle-like trajector centered at the origin when x is small, and, respectivel, a square-like trajector when x = O(), which clearl shows the effect of the boundar. This is quite different from the situation in bounded domains with the Dirichlet BC where the motion of a single vortex depends significantl on its initial location for some h(x). It is also quite different from the situation in the whole space where a single vortex does not move at all under the initial condition (5.) when Ω = R 2. (ii) As, the dnamics of the vortex center under the NLSE dnamics converges uniforml in time to that of the vortex center under the reduced dnamical laws. In fact, based on our extensive numerical experiments, the motion of the vortex center from the reduced dnamical laws agrees with that of the vortex centers from the NLSE dnamics qualitativel when <<and quantitativel when < Vortex pair. Here we present numerical results of the interactions of a vortex pair under the NLSE (.) dnamics and its corresponding reduced dnamical laws; i.e., we take M =2,n = n 2 =,andx 2 = x =(d, ) with <d < in (5.). Figure 5 depicts the trajector of the vortex pair, time evolution of E (t), Ekin (t), x (t), x 2(t), and d (t) when = 32 in (.) and d =.5 in(5.). From Figure 5 and additional numerical results (not shown here for brevit), we can draw the following conclusions for the interactions of a vortex pair under the NLSE dnamics (.) with the homogeneous Neumann BC: (i) The total energ is conserved numericall ver well during the dnamics. (ii) The two vortices in the vortex pair move periodicall along a circle-like trajector centered at the origin when d is small, and, respectivel, a square-like trajector when d = O(), which clearl show the effect of the boundar. In addition, the trajectories are smmetric. (iii) When, the dnamics of the two vortex centers under the NLSE dnamics converges uniforml in time to that of two vortex centers under the reduced dnamical laws. This verifies numericall the validation of the reduced dnamical laws in this case. In fact, based on our extensive numerical experiments, the motion of the two vortex centers from the reduced dnamical laws agrees with with that of the two vortex centers from the NLSE dnamics qualitativel when <<and quantitativel when < Vortex dipole. Here we present numerical results of the interactions of a vortex dipole under the NLSE (.) dnamics and its corresponding reduced dnamical laws; i.e., we take M =2,n = n 2 =, x 2 = x =(d, ) with different d,

23 QUANTIZED VORTEX IN NLSE ON BOUNDED DOMAINS 433 x or x2 t 2 or 2 x or x2 t 2 or 2 x x or x2 t 2.65 t.3 x.8 =/6 d.9 =/25 =/32 t 2.4 =/ d.7 =/2 =/4 x or 2.65 t.3 t 2 t 2 Fig. 6. Trajector and time evolution of x (t) and x 2 (t) for d =.25 (top left two figures), d =.7 (top right two figures), and d =. (bottom left two figures), and time evolution of d (t) for d =.25 and d =.7 (bottom right two figures) in section 5.3. and = 32 in (.). Figure 6 depicts the trajector of the vortex dipole, and time evolution of x (t), x 2 (t), and d (t). From Figure 6 and additional numerical results (not shown here for brevit), we can draw the following conclusions for the interactions of a vortex dipole under the NLSE dnamics (.) with the homogeneous Neumann BC: (i) The total energ is conserved numericall ver well during the dnamics. (ii) The pattern of the motion of the two vortices depends on their initial locations. (iii) The two vortices will move smmetricall (and periodicall if the are well separated) with respect to the -axis. Moreover, there exists a critical value d r c = d c = d c for <<, which is found numericall as d c =.5, such that if initiall d <d c, the two vortices will move first upward toward the top boundar, then turn outward to the side boundar, and finall move counterclockwise and clockwise, respectivel (cf. Figure 6). While if d >d c, then the will move first downward toward the bottom boundar, then turn inward to the domain, and finall move counterclockwise and clockwise, respectivel (cf. Figure 6). Certainl, when d =.5, the vortex dipole does not move due to smmetr. (iv) For fixed <<, there exists another critical value ˆd c satisfing lim ˆd c = such that if d < ˆd c, the two vortices in the dipole under the NLSE dnamics will merge at a finite time T c depending on and d (cf. Figure 6). However, the two vortices in the dipole from the reduced dnamical laws never merge at finite time. Hence, the reduced dnamical laws fail qualitativel if <d < ˆd. (v) For fixed d,when, the dnamics of the two vortex centers in the NLSE dnamics converges uniforml in time to that of the two vortex centers in the reduced dnamical laws before the two vortices merge with each other (cf. Figure 6), which verifies numericall the validation of the reduced dnamical laws in this case. In fact, based on our extensive numerical experiments, the motion of the two vortex centers from the reduced dnamical laws agrees with that of the two vortex centers in the NLSE dnamics qualitativel when << and quantitativel when < Vortex cluster. Here we present numerical studies on the dnamics of vortex clusters in the NLSE (.) with homogeneous Neumann BC; i.e., we choose

24 434 WEIZHU BAO AND QINGLIN TANG.4 x x.4.4 x.4.6 x 3 (t) 3 (t).6 x 3 (t) 3 (t).6 x 3 (t) 3 (t).3 x (t) 2 2 (t).3 x 2 (t) 2 (t).3 x 2 (t) (t) t t t.3.45 Fig. 7. Trajector of the vortex x (solid line), x 2 (dashed-dotted line), and x 3 (dashed line) and their corresponding time evolution for Case I during small time (left column), intermediate time(middlecolumn),andlargetime(rightcolumn)with = 4 and d =.25 in section 5.4. the initial data (.2) as (5.) and stud the following four cases: Case I. M =3,n = n 2 = n 3 =,x = x 3 =(d, ), and x 2 =(, ). Case II. M =4,n = n 2 = n 3 = n 4 =,x = x 2 =(d, ), and x 3 = x 4 = (d 2, ) with <d d 2 <. Case III. M = 4, n = n 2 = n 3 = n 4 =, x = x 2 = (d, ), and x 3 = x 4 =(,d 2)with<d,d 2 <. Case IV. M =9,n = = n 9 =, and the vortex centers are initiall located on the 3 3 uniform mesh points for the rectangle [ d,d ] 2 with <d <. Figure 7 shows trajector and time evolution of x (t), x 2(t), and x 3(t) inthe NLSE dnamics for Case I. Figure 8 depicts contour plots of ψ at different times in NLSE dnamics for Cases II and III, and Figure 9 shows contour plots of ψ and slice plots of ψ(,,t) in NLSE dnamics for Case IV showing sound wave propagation. Based on Figures 7 9 and additional results (not shown here for brevit), we can draw the following conclusions: (i) For Case I, the dnamics of the vortices is quite similar to that of the vortices under the Dirichlet BC in section 4.4 and that of the vortices in the trapped BEC [8, 3, 4]. The middle vortex (initiall at the origin) will not move, while the other two vortices rotate clockwise around the origin for some time. This dnamics agrees ver well with the NLSE dnamics in the whole plane [43, 44]. After some time, the above smmetric structure is broken due to boundar effect and numerical errors; i.e., the middle vortex will begin to move toward one of the other two vortices and form a pair of vortices, then the two vortices in the pair will rotate with each other and the vortex pair will rotate with the leftover single vortex for a while. Then this pair will separate and one of them will form a new vortex pair with the single vortex, leaving the other to be a new single vortex rotating with the new vortex pair. This process will be repeated tautologicall like three dancers exchanging their partners alternativel. In fact, the dnamical pattern after the smmetric structure breaking depends highl on the mesh size and time step. (ii) For Case II, the four vortices form as two vortex pairs when t is small. These two pairs rotate with each other clockwise; meanwhile, the two vortices in each pair

25 QUANTIZED VORTEX IN NLSE ON BOUNDED DOMAINS 435 Fig. 8. Contour plots of ψ (x,t) with = at different times for the NLSE dnamics of a 6 vortex cluster for Case II with d =.6,d 2 =.3 (top two rows) and Case III with d = d 2 =.3 (bottom two rows) in section 5.4. also rotate with each other clockwise, and radiations and sound waves are emitted. The sound waves propagate outward and are reflected back into the domain when the hit the boundar, which push the two vortex pairs closer. When the two vortex pairs get close enough, the two vortices with the smallest distance among the four form a new vortex pair and leave the remaining two as single vortices. The vortex pair rotates around the origin. This process is iterativel repeated during the dnamics (cf. Figure 8, top two rows). (iii) For Case III, when t is small, the four vortices form two vortex dipoles and move smmetricall with respect to the line = x toward the top right and bottom left corners, respectivel. Meanwhile, the two vortices in each dipole move smmetricall with respect to the line = x. After a while, and when the two dipoles arrive at some place near the corners, the two vortices in each dipole split from each other and reformulate two different dipoles. After this, the two vortices in each dipole move smmetricall with respect to the line = x, and the two new dipoles then

26 436 WEIZHU BAO AND QINGLIN TANG.6 t=.2 t=.6 ψ (,,t).53 ψ (,,t).56 ψ (,,t) t= ψ (,,t) t= ψ (,,t). t= ψ (,,t).8 t=.5.54 ψ (,,t).8 t=.7.59 ψ (,,t).8 t= Fig. 9. Contour plots of ψ (x,t) (left) and slice plots of ψ (,,t) (right) at different times under the NLSE dnamics of a vortex cluster in Case IV with d =.5 and = 4 showing sound wave propagation in section 5.4. move smmetricall with respect to the line = x toward their initial locations. This process is then repeated periodicall (cf. Figure 8, bottom two rows). (iv) For Case IV, the vortex initiall centered at the origin does not move due to smmetr, and the other eight vortices rotate clockwise and move along two circlelike trajectories (cf. Figure 9). During the dnamics, sound waves are generated and propagate outward. Some of the sound waves will exit out of the domain, while others are reflected back into the domain when the hit the boundar. The distances between the one located at the origin and the other eight vortices become larger when the sound waves are radiated outward, while the decrease when the sound waves are reflected from the boundar and move back into the domain (cf. Figure 9). Again, similarl to the case in BEC setups [2, 28, 36, 42], the smmetric structure near t = in Case I for three vortices is dnamicall unstable, i.e., smmetric structure breaking will happen ver quickl if we perturb the location of either the central vortex or one of the side vortices a little bit Radiation and sound wave. Here we stud numericall how the radiation and sound waves affect the dnamics of quantized vortices in the NLSE dnamics under the homogeneous Neumann BC. To this end, we take the initial data (.2) as (4.4) with ψ chosen as (5.) with M =2,n = n 2 =,andx = x 2 =(., ); i.e., we perturb the initial data for studing the interactions of a vortex pair b a Gaussian function with amplitude δ. Then we take δ =, let go to, and solve